Geographic Information Science
Geography 625
Intermediate
Geographic Information Science
Week 10_11: Area objects and spatial autocorrelation
Instructor: Changshan Wu
Department of Geography
The University of Wisconsin-Milwaukee
Fall 2006
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Outline
1. Introduction
2. Geometric properties of areas
3. Spatial autocorrelation: joins count approach
4. Spatial autocorrelation: Moran’s I
5. Spatial autocorrelation: Geary’s C
6. Spatial autocorrelation: weight matrices
7. Local indicators of spatial association (LISA)
8. Spatial regression
9. Spatial expansion method
10. Geographical weighted regression
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1. Introduction
- Types of area object
1. Natural areas: self-defining, their boundaries are defined
by the phenomenon itself (e.g. lake, land use)
Lake map
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1. Introduction
- Types of area object
2. Imposed areas: area objects are imposed by human beings,
such as countries, states, counties etc. Boundaries are defined
independently of any phenomenon, and attribute values are
enumerated by surveys or censuses.
Potential Problems
1.
2.
3.
may bear little relationship to underlying patterns
Arbitrary and modifiable
Danger of ecological fallacies (aggregated format)
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1. Introduction
- Types of area object
3. Raster: space is divided into small regular grid cells.



In a raster, the area objects are identical and together cover the region of
interest.
Each cell can be considered an area object.
Raster data are always used to represent continuous phenomenon.
Squares
Hexagons
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1. Introduction
- Types of area object
Planar enforced: area objects mesh together neatly and exhaust
the study region, so that there is no holes, and every location is
inside just a single area.
Not planar enforced: the areas do not fill or exhaust the space,
the entities are isolated from one another, or perhaps overlapped
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2. Geometric Properties of Areas
- Area
y
(x3, y3)
Area 
(x4, y4)
(x2, y2)
n
1
2
 (x
i 1
i 1
 xi )( yi 1  yi )
Assume x1 = xn+1
(x1, y1)
x
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2. Geometric Properties of Areas
- Skeleton
The skeleton of a polygon is
a network of lines inside a
polygon constructed so that
each point on the network is
equidistant from the nearest
two edges in the polygon
boundary.
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2. Geometric Properties of Areas
- Skeleton
Center derived by skeleton analysis
n
center
Arithmetic
center
xˆ   xi
i 1
n
yˆ   yi
i 1
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2. Geometric Properties of Areas
- Shape
Shape: a set of relationships of relative position between points
on their perimeters.
 In ecology, the shapes of patches of a specified habitat are
thought to have significant effects on what happens and
around them.
 In urban studies, urban shapes change from traditional
monocentric to polycentric sprawl
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2. Geometric Properties of Areas
- Shape
Perimeter: P
Area: a
Longest axis: L1
Second axis: L2
The radius of the largest internal
circle: R1
The radius of the smallest
enclosing circle: R2
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2. Geometric Properties of Areas
- Shape
Compactness ratio
 a / a2
a is the area of the polygon
a2 is the area of the circle having the same perimeter (P) as
the object
What is the compactness ratio for a circle?
What is the compactness ratio for a square?
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2. Geometric Properties of Areas
- Shape
Other measurements
Elongation ratio: L1/L2
Form ratio:
2
1
a/L
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3. Spatial autocorrelation
- Joins count approach
Developed by Cliff and Ord (1973) in their book: Spatial
Autocorrelation
The joins count statistic is applied to a map of areal units
where each unit is classified as either black (B) or white
(W).
The joins count is determined by counting the number of
occurrences in the map of each of the possible joins (e.g.
BB, WW, BW) between neighboring areal units.
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3. Spatial autocorrelation
- Joins count approach
Possible joins:
JBB: the number of joins of BB
JWW: the number of joins of WW
JBW: the number of joins of BW or WB
Neighbor definition
Rook’s case: four neighbors (North-South-West-East)
Queen’s case: eight neighbors (including diagonal neighbors)
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3. Spatial autocorrelation
- Joins count approach
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3. Spatial autocorrelation
- Joins count approach
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3. Spatial autocorrelation
- Joins count approach
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3. Spatial autocorrelation
- Joins count approach
Statistical tests for spatial correlation
Independent Random Process (IRP)
Mean
E ( J BB )  kpB2
E ( JW W )  kp
2
W
E ( J BW )  2kpB pw
Where k is the total number of joins on the map
pB is the probability of an area being coded B
pW is the probability of an area being coded W
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3. Spatial autocorrelation
- Joins count approach
Independent Random Process (IRP)
The expected standard deviations are as follows
E ( sBB )  kpB2  2mpB3  (k  2m) pB4
E ( sW W )  kpW2  3mpW3  (k  2m) pW4
E ( sBW )  2(k  m) pB pW  4(k  2m) pB2 pW2
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3. Spatial autocorrelation
- Joins count approach
Independent Random Process (IRP)
1 n
m   ki (ki  1)
2 i 1
ki is the number of joins to the ith area
m = 0.5 [(4×2×1) + (16×3×2)+(16×4×3)]
center
corners
edges
= 148
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3. Spatial autocorrelation
- Joins count approach
Independent Random Process (IRP)
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3. Spatial autocorrelation
- Joins count approach
Independent Random Process (IRP)
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3. Spatial autocorrelation
- Joins count approach
Z BB
J BB  E ( J BB )

E ( sBB )
Z BW
J BW  E ( J BW )

E ( sBW )
ZW W
JW W  E ( JW W )

E ( sW W )
A large negative Z-score on
JBW indicates positive
autocorrelation since it
indicates that there are fewer
BW joins than expected.
A large positive Z-score on JBW
is indicative of negative
autocorrelation.
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3. Spatial autocorrelation
- Joins count approach
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3. Spatial autocorrelation
- Joins count approach (example)
B: Bush
W: Gore
State-level results for the 2000 U.S. presidential election
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3. Spatial autocorrelation
- Joins count approach (example)
Adjacency (joins)
matrix: if two states
share a common
boundary, they are
adjacent.
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3. Spatial autocorrelation
- Joins count approach (example)
Bush
48,021,500
pB = 0.49885
Gore
48,242,921
pw = 0.50115
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4. Spatial autocorrelation
- Moran’s I
Limitations of Joins Count Statistics
1. It can only be applied on binary data
2. Although the approach provides an indication of the strength
of autocorrelation present in terms of Z-scores, it is not
readily interpreted, particularly if the results of different tests
appear contradictory
3. The equations for the expected values of counts are fairly
formidable.
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4. Spatial autocorrelation
- Moran’s I
n
I
 w ( y
n
n
(y
i 1
i
n
i 1 j 1
 y)
ij
n
2
i
 y )( y j  y )
n
 w
i 1 j 1
ij
1
If zone i an zone j are adjacent
0
otherwise
wij =
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4. Spatial autocorrelation
- Moran’s I
n
1
2
2
2
0
0
3
4
1
2
A= 3
4
1
0
1
1
0
I
n
n
(y
i 1
2
1
0
0
1
3
1
0
0
1
2

y
)
i
n
 w ( y
i 1 j 1
ij
n
i
 y )( y j  y )
n
 w
i 1 j 1
ij
4
0
1
1
0
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4. Spatial autocorrelation
- Moran’s I
1
2
0
2
0
2
3
n
I
1
2
A= 3
4
(y
i 1
4
1
0
1
1
0
2
1
0
0
1
3
1
0
0
1
 w ( y
n
n
i
 y)
n
i 1 j 1
ij
n
2
i
 y )( y j  y )
n
 w
i 1 j 1
ij
4
0
1
1
0
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4. Spatial autocorrelation
- Moran’s I
For Moran’s I, a positive value indicates a positive
autocorrelation, and a negative value indicates a negative
autocorrelation.
Moran’s I is not strictly in the range of -1 to +1.
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5. Spatial autocorrelation
- Geary’s C
Proposed by Geary’s contiguity ratio C
n
C
n 1
n
2
(
y

y
)
 i
i 1
n
2
w
(
y

y
)
 ij i j
i 1 j 1
n
n
2 wij
i 1 j 1
1
If zone i an zone j are adjacent
0
otherwise
wij =
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5. Spatial autocorrelation
- Geary’s C
1
2
0
2
0
2
3
n
C
4
1
2
A= 3
4
(y
i 1
1
0
1
1
0
2
1
0
0
1
3
1
0
0
1
 w ( y
n 1
n
i
 y)
n
ij
i 1 j 1
n
2
i
 y j )2
n
2 wij
i 1 j 1
4
0
1
1
0
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5. Spatial autocorrelation
- Geary’s C
The value generally varies between 0 and 2.
The theoretical value of C is 1 under independent random
process. values less than 1 indicate positive spatial
autocorrelation while values greater than 1 indicate negative
autocorrelation.
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6. Spatial autocorrelation
- Other Weighting Matrices
1. Using distance
d ijz
wij  
0
Where dij < D and z < 0
Where dij > D
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6. Spatial autocorrelation
- Other Weighting Matrices
2. Using the length of shared boundary
wij 
lij
li
Where li is the length of the boundary of zone i
lij is the length of boundary shared by area i and j
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6. Spatial autocorrelation
- Other Weighting Matrices
3. Using both distance and the length of shared boundary
wij 
z
ij ij
d l
li
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7. Spatial autocorrelation
- Local Indicators
Global statistics tell us whether or not an overall configuration
is autocorrelated, but not where the unusual interactions are.
Local indicators of spatial association (LISA) were proposed
in Getis and Ord (1992) and Anselin (1995).
These are disaggregate measures of autocorrelation that
describe the extent to which particular areal units are similar
to, or different from, their neighbors.
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7. Spatial autocorrelation
- Local Indicators
Local Gi
Gi



j i
n
wij yi
i 1
yi
Used to detect possible nonstationarity in data, when clusters
of similar values are found in specific subregions of the area
studied.
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7. Spatial autocorrelation
- Local Indicators
1
2
0
2
0
2
3
4
Gi



j i
n
wij yi
i 1
yi
1 If zone i an zone j are adjacent
wij =
0
otherwise
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7. Spatial autocorrelation
- Local Indicators
Local Moran’s I
I i  zi  wij z j
j i
Where
zi  ( yi  y ) / s
W matrix is a row-standardized (i.e. scaled
so that each row sums to 1)
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7. Spatial autocorrelation
- Local Indicators
1
2
0
2
0
2
I i  zi  wij z j
3
4
zi  ( yi  y ) / s
1
2
3
4
1
0
1
1
0
j i
2
1
0
0
1
3
1
0
0
1
4
0
1 =
1
0
0 1/ 2 1/ 2 0
1/ 2 0
0 1/ 2
1/ 2 0 0 1/ 2
0 1/ 2 1/ 2 0
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7. Spatial autocorrelation
- Local Indicators
Local Geary’s C
Ci   wij ( yi  y j )
1
2
0
2
0
2
3
4
2
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8. Spatial regression Models
Y  X  U
Where U is a zero-mean vector of errors with variancecovariance matrix C
E (U )  0
E (UU )  C
T
If C = I, this is the ordinary least
square (OLS) model
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8. Spatial regression Models
- Simultaneous autocorrelation model (SAR)
Y  X  U
U  WU  
Y  X  WU  
 X  W (Y  X )  
 X  WY  WX  
and
E ( )  0
E ( T )   2 I
C  E (UU T )
  2 ((1  W )T (1  W )) 1
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8. Spatial regression Models
- Simultaneous autocorrelation model (SAR)
Software for SAR model
ArcView 3.2 + S-Plus
R programming language
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9. Spatial Expansion Models
Proposed by Casetti (1972)
OLS model
Expansion
model
Y  b0  b1 x1  b2 x2  ...  bn xn  
Y  b0  b1 ( Z ) x1  b2 ( Z ) x2  ...  bn ( Z ) xn  
b1 ( Z )  c0  c1 z1  c2 z 2  ...
...
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10. Geographic Weighted Regression
OLS model
Y  b0  b1 x1  b2 x2  ...  bn xn  
GWR model
Yi  b0i  b1i x1i  b2i x2i  ...  bni xni   i
...
The coefficients b vary with respect to the location i
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10. Geographic Weighted Regression
Software
R program
GWR package (free from Fotheringham)
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